改进型扩展比例边界有限元法
|
摘要
结合了扩展有限元法(extended finite element methods,XFEM)和比例边界有限元法(scaled boundary finite element methods,SBFEM)的主要优点,提出了一种改进型扩展比例边界有限元法(improved extended scaled boundary finite element methods,i XSBFEM),为断裂问题模拟提供了一条新的途径.类似XFEM,采用两个正交的水平集函数表征材料内部裂纹面,并基于水平集函数判断单元切割类型;将被裂纹切割的单元作为SBFE的子域处理,采用SBFEM求解单元刚度矩阵,从而避免了XFEM中求解不连续单元刚度矩阵需要进一步进行单元子划分的缺陷;同时,借助XFEM的主要思想,将裂纹与单元边界交点的真实位移作为单元结点的附加自由度考虑,赋予了单元结点附加自由度明确的物理意义,可以直接根据位移求解结果得出裂纹与单元边界交点的位移;对于含有裂尖的单元,选取围绕裂尖单元一圈的若干层单元作为超级单元,并将此超级单元作为SBFE的一个子域求解刚度矩阵,超级单元内部的结点位移可通过SBFE的位移模式求解得到,应力强度因子可基于裂尖处的奇异位移(应力)直接获得,无需借助其他的数值方法.最后,通过若干数值算例验证了建议的i XSBFEM的有效性,相比于常规XFEM,i XSBFEM的基于位移范数的相对误差收敛性较好;采用i XSBFEM通过应力法和位移法直接计算得到的裂尖应力强度因子均与解析解吻合较好.
Combining the main advantages of the extended finite element methods(XFEM) and the scaled boundary finite element methods(SBFEM), improved extended scaled boundary finite element methods(i XSBFEM) are proposed.The proposed methods can provide a new way for the simulation of fracture problems. Similar to XFEM, two orthogonal level set functions are used to characterize the internal crack surface in materials, and how the element is partitioned by a crack can be judged by level set functions. These elements partitioned by the crack are treated as a subdomain of SBFEM, and then the element stiffness matrix of these discontinuous elements can be directly solved by SBFEM, thus avoiding the need for further element subdivision to the solution of the discontinuous element stiffness matrix in XFEM.At the same time, with the help of the main idea of XFEM, the real displacement of the intersection point between the crack and the element boundary is considered as the additional degrees of freedom of the element nodes, thereby it gives explicit physical meaning of additional degrees of freedom. For the element containing the crack tip, several layers of elements around the crack tip are selected as a super element, and the super element is used as a subdomain of SBFE to solve the stiffness matrix. The node displacement inside the super element can be obtained by the SBFE displacement approximation. The stress intensity factor can be directly obtained based on the singular displacement or stress at the crack tip, without the need of other numerical methods. Finally, several numerical examples are given to verify the effectiveness of the proposed i XSBFEM. Compared with the standard XFEM, the relative error convergence of the i XSBFEM based on the displacement norm is better, and the stress intensity factors computed by stress based and displacement based method in i XSBFEM both are in good agreement with the analytical solution.
Combining the main advantages of the extended finite element methods(XFEM) and the scaled boundary finite element methods(SBFEM), improved extended scaled boundary finite element methods(i XSBFEM) are proposed.The proposed methods can provide a new way for the simulation of fracture problems. Similar to XFEM, two orthogonal level set functions are used to characterize the internal crack surface in materials, and how the element is partitioned by a crack can be judged by level set functions. These elements partitioned by the crack are treated as a subdomain of SBFEM, and then the element stiffness matrix of these discontinuous elements can be directly solved by SBFEM, thus avoiding the need for further element subdivision to the solution of the discontinuous element stiffness matrix in XFEM.At the same time, with the help of the main idea of XFEM, the real displacement of the intersection point between the crack and the element boundary is considered as the additional degrees of freedom of the element nodes, thereby it gives explicit physical meaning of additional degrees of freedom. For the element containing the crack tip, several layers of elements around the crack tip are selected as a super element, and the super element is used as a subdomain of SBFE to solve the stiffness matrix. The node displacement inside the super element can be obtained by the SBFE displacement approximation. The stress intensity factor can be directly obtained based on the singular displacement or stress at the crack tip, without the need of other numerical methods. Finally, several numerical examples are given to verify the effectiveness of the proposed i XSBFEM. Compared with the standard XFEM, the relative error convergence of the i XSBFEM based on the displacement norm is better, and the stress intensity factors computed by stress based and displacement based method in i XSBFEM both are in good agreement with the analytical solution.
引文
1 Belytschko T,Black T.Elastic crack growth in finite elements with minimal remeshing.International Journal for Numerical Methods in Engineering,1999,45(5):601-620
2 Xu DD,Liu ZL,Zhuang Z.Recent advances in the extended finite element method(XFEM)and isogeometric analysis(IGA)[N].2016,59(12):124631
3 Wang T,Liu ZL,Zeng QL,et al.XFEM modeling of hydraulic fracture in porous rocks with natural fractures.Science China Physics Mechanics&Astronomy,2017,60(8):084612
4石杰,李庆斌.基于扩展有限元的重力坝尺寸效应.清华大学学报(自然科学版),2017,57(4):345-350(Shi Jie,Li Qingbin.Size effects of concrete gravity dams based on XFEM analysis.Journal of Tsinghua University(Science&Technology),2017,57(4):345-350(in Chinese))
5 Moёs N,Dolbow J,Belytschko T.A finite element method for crack growth without remeshing.International Journal for Numerical Methods in Engineering,1999,46(1):131-150
6江守燕,杜成斌.弱不连续问题扩展有限元法的数值精度研究.力学学报,2012,44(6):1005-1015(Jiang Shouyan,Du Chengbin.Study on numerical precision of extended finite element methods for modeling weak discontinuties.Chinese Journal of Theoretical and Applied Mechanics,2012,44(6):1005-1015(in Chinese))
7 Bordas SP,Rabczuk T,Hung NX et al.Strain smoothing in FEMand XFEM.Computers&Structures,2010,88(23-24):1419-1443
8 Ventura G.On the elimination of quadrature subcells for discontinuous functions in the eXtended Finite-Element Method.International Journal for Numerical Methods in Engineering,2006,66(5):761-779
9 Natarajan S,Mahapatra DR,Bordas SP.Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework.International Journal for Numerical Methods in Engineering,2010,83(3):269-294
10 Natarajan S,Bordas SP,Mahapatra D.Numerical integration over arbitrary polygonal domains based on Schwarz-Christoffel conformal mapping.International Journal for Numerical Methods in Engineering,2009,80(1):103-134
11田荣,文龙飞.改进型XFEM进展.计算力学学报,2016,33(4):469-477(Tian Rong,Wen Longfei.Recent progresses on improved XFEM.Chinese Journal of Computational Mechanics,2016,33(4):469-477(in Chinese))
12文龙飞,王理想,田荣.动载下裂纹应力强度因子计算的改进型扩展有限元法.力学学报,2018,50(3):599-610(Wen Longfei,Wang Lixiang,Tian Rong.Accrate computation on dynamic SIFs using improved XFEM.Chinese Journal of Theoretical and Applied Mechanics,2018,50(3):599-610(in Chinese))
13江守燕,杜成斌,顾冲时等.求解双材料界面裂纹应力强度因子的扩展有限元法.工程力学,2015,32(3):22-27(Jiang Shouyan,Du Chengbin,Gu Chongshi,et al.Computation of stress intensity factors for interface cracks between two dissimilar materials using extended finite element methods.Engineering Mechanics,2015,32(3):22-27(in Chinese))
14江守燕,杜成斌.动载下缝端应力强度因子计算的扩展有限元法.应用数学和力学,2013,34(6):586-597(Jiang Shouyan,Du Chengbin.Evaluation on stress intensity factors at the crack tip under dynamic loads using extended finite element methods.Applied Mathematics and Mechanics,2013,34(6):586-597(in Chinese))
15刘学聪,章青,夏晓舟.一种新型裂尖加强函数的显式动态扩展有限元法.工程力学,2017,34(10):10-18(Liu Xuecong,Zhang Qing,Xia Xiaozhou.A new enrichment function of crack tip in XFEM dynamics by explicit time algorithm.Engineering Mechanics,2017,34(10):10-18(in Chinese))
16 Song C,Wolf JP.The scaled boundary finite-element method-alias consistent infinitesimal finite-element cell method-for elastodynamics.Computer Methods in Applied Mechanics and Engineering,1997,147(3):329-355
17 Deeks AJ,Wolf JP.A virtual work derivation of the scaled boundary finite-element method for elastostatics.Computational Mechanics,2002,28(6):489-504
18 Song C,Wolf JP.Semi-analytical representation of stress singularities as occurring in cracks in anisotropic multi-materials with the scaled boundary finite-element method.Computers and Structures,2002,80(2):183-197
19 Huang YJ,Yang ZJ,Liu GH,et al.An efficient FE-SBFE coupled method for mesoscale cohesive fracture modelling of concrete.Computational Mechanics,2016,58(4):635-655
20钟红,暴艳利,林皋.基于多边形比例边界有限元的重力坝裂缝扩展过程模拟.水利学报,2014,45(S1):30-37(Zhong Hong,Bao Yanli,Lin Gao.Modelling of crack propagation of gravity dams based on scaled boundary polygons.Journal of Hydraulic Engineering,2014,45(S1):30-37(in Chinese))
21陈楷,邹德高,孔宪京等.多边形比例边界有限单元非线性化方法及应用.浙江大学学报(工学版),2017,51(10):1996-2004(Chen Kai,Zou Degao,Kong Xianjing,et al.Novel nonlinear polygon scaled boundary finite element method and its application.Journal of Zhejiang University(Engineering Science),2017,51(10):1996-2004(in Chinese))
22章鹏,杜成斌,江守燕.比例边界有限元法求解裂纹面接触问题.力学学报,2017,49(6):1335-1347(Zhang Peng,Du Chengbin,Jiang Shouyan.Crack face contact problem analysis using the scaled boundary finite element method.Chinese Journal of Theoretical and Applied Mechanics,2017,49(6):1335-1347(in Chinese))
23 Natarajan S,Song C.Representation of singular fields without asymptotic enrichment in the extended finite element method.International Journal for Numerical Methods in Engineering,2013,96(13):813-841
24陈白斌,李建波,林皋.基于X-SBFEM的裂纹体非网格重剖分耦合模型研究.工程力学,2015,32(3):15-21(Chen Baibin,Li Jianbo,Lin Gao.Study on the coupling model of crack without remeshing based on X-SBFEM.Engineering Mechanics,2015,32(3):15-21(in Chinese))
25陈白斌,李建波,林皋.无需裂尖增强函数的扩展比例边界有限元法.水利学报,2015,46(4):489-496(Chen Baibin,Li Jianbo,Lin Gao.An extended scaled boundary finite element method without asymptotic enrichment of the crack tip.Journal of Hydraulic Engineering,2015,46(4):489-496,504(in Chinese))
26李建波,陈白斌,林皋.基于水平集算法的扩展比例边界有限元法研究.工程力学,2016,33(8):8-14(Li Jianbo,Chen Baibin,Lin Gao.Study on the extended scaled boundary finite element method based on level set algorithm.Engineering Mechanics,2016,33(8):8-14(in Chinese))
27傅兴安,李建波,林皋.基于X-SBFEM的非线性断裂数值模型研究.大连理工大学学报,2017,57(5):494-500(Fu Xing’an,Li Jianbo,Lin Gao.Study of numerical model of nonlinear fracture based on X-SBFEM.Journal of Dalian University of Technology,2017,57(5):494-500(in Chinese))
28 Jiang S,Du C.Coupled finite volume methods and extended finite element methods for the dynamic crack propagation modelling with the pressurized crack surfaces.Shock and Vibration,2017,2017:3751340
29 Jiang S,Du C.Study on dynamic interaction between crack and inclusion or void by using XFEM.Structural Engineering&Mechanics,2017,63(3):329-345
30 Fries TP.A corrected XFEM approximation without problems in blending elements.International Journal for Numerical Methods in Engineering,2008,75:503-532
31江守燕,杜成斌.一种XFEM断裂分析的裂尖单元新型改进函数.力学学报,2013,45(1):134-138(Jiang Shouyan,Du Chengbin.A novel enriched function of elements containing crack tip for fracture analysis in XFEM.Chinese Journal of Theoretical and Applied Mechanics,2013,45(1):134-138(in Chinese))
32 Pommier S,Gravouil A,Combescure A,et al.Extended Finite Element Method for Crack Propagation.ISTE Ltd and John Wiley&Sons,Inc.,2011:8-9
33 Mohammadi S.Extended Finite Element Method.Blackwell Publishing Ltd,2008:28-29
34 Sih GC,Paris PC,Erdogan F.Crack-tip stress intensity factors for plane extension and plane bending problems.Journal of Applied Mechanics,1962,29:306-312
35 Zamani A,Gracie R,Eslami MR.Cohesive and non-cohesive fracture by higher-order enrichment of XFEM.International Journal for Numerical Methods in Engineering,2012,90:452-483
2 Xu DD,Liu ZL,Zhuang Z.Recent advances in the extended finite element method(XFEM)and isogeometric analysis(IGA)[N].2016,59(12):124631
3 Wang T,Liu ZL,Zeng QL,et al.XFEM modeling of hydraulic fracture in porous rocks with natural fractures.Science China Physics Mechanics&Astronomy,2017,60(8):084612
4石杰,李庆斌.基于扩展有限元的重力坝尺寸效应.清华大学学报(自然科学版),2017,57(4):345-350(Shi Jie,Li Qingbin.Size effects of concrete gravity dams based on XFEM analysis.Journal of Tsinghua University(Science&Technology),2017,57(4):345-350(in Chinese))
5 Moёs N,Dolbow J,Belytschko T.A finite element method for crack growth without remeshing.International Journal for Numerical Methods in Engineering,1999,46(1):131-150
6江守燕,杜成斌.弱不连续问题扩展有限元法的数值精度研究.力学学报,2012,44(6):1005-1015(Jiang Shouyan,Du Chengbin.Study on numerical precision of extended finite element methods for modeling weak discontinuties.Chinese Journal of Theoretical and Applied Mechanics,2012,44(6):1005-1015(in Chinese))
7 Bordas SP,Rabczuk T,Hung NX et al.Strain smoothing in FEMand XFEM.Computers&Structures,2010,88(23-24):1419-1443
8 Ventura G.On the elimination of quadrature subcells for discontinuous functions in the eXtended Finite-Element Method.International Journal for Numerical Methods in Engineering,2006,66(5):761-779
9 Natarajan S,Mahapatra DR,Bordas SP.Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework.International Journal for Numerical Methods in Engineering,2010,83(3):269-294
10 Natarajan S,Bordas SP,Mahapatra D.Numerical integration over arbitrary polygonal domains based on Schwarz-Christoffel conformal mapping.International Journal for Numerical Methods in Engineering,2009,80(1):103-134
11田荣,文龙飞.改进型XFEM进展.计算力学学报,2016,33(4):469-477(Tian Rong,Wen Longfei.Recent progresses on improved XFEM.Chinese Journal of Computational Mechanics,2016,33(4):469-477(in Chinese))
12文龙飞,王理想,田荣.动载下裂纹应力强度因子计算的改进型扩展有限元法.力学学报,2018,50(3):599-610(Wen Longfei,Wang Lixiang,Tian Rong.Accrate computation on dynamic SIFs using improved XFEM.Chinese Journal of Theoretical and Applied Mechanics,2018,50(3):599-610(in Chinese))
13江守燕,杜成斌,顾冲时等.求解双材料界面裂纹应力强度因子的扩展有限元法.工程力学,2015,32(3):22-27(Jiang Shouyan,Du Chengbin,Gu Chongshi,et al.Computation of stress intensity factors for interface cracks between two dissimilar materials using extended finite element methods.Engineering Mechanics,2015,32(3):22-27(in Chinese))
14江守燕,杜成斌.动载下缝端应力强度因子计算的扩展有限元法.应用数学和力学,2013,34(6):586-597(Jiang Shouyan,Du Chengbin.Evaluation on stress intensity factors at the crack tip under dynamic loads using extended finite element methods.Applied Mathematics and Mechanics,2013,34(6):586-597(in Chinese))
15刘学聪,章青,夏晓舟.一种新型裂尖加强函数的显式动态扩展有限元法.工程力学,2017,34(10):10-18(Liu Xuecong,Zhang Qing,Xia Xiaozhou.A new enrichment function of crack tip in XFEM dynamics by explicit time algorithm.Engineering Mechanics,2017,34(10):10-18(in Chinese))
16 Song C,Wolf JP.The scaled boundary finite-element method-alias consistent infinitesimal finite-element cell method-for elastodynamics.Computer Methods in Applied Mechanics and Engineering,1997,147(3):329-355
17 Deeks AJ,Wolf JP.A virtual work derivation of the scaled boundary finite-element method for elastostatics.Computational Mechanics,2002,28(6):489-504
18 Song C,Wolf JP.Semi-analytical representation of stress singularities as occurring in cracks in anisotropic multi-materials with the scaled boundary finite-element method.Computers and Structures,2002,80(2):183-197
19 Huang YJ,Yang ZJ,Liu GH,et al.An efficient FE-SBFE coupled method for mesoscale cohesive fracture modelling of concrete.Computational Mechanics,2016,58(4):635-655
20钟红,暴艳利,林皋.基于多边形比例边界有限元的重力坝裂缝扩展过程模拟.水利学报,2014,45(S1):30-37(Zhong Hong,Bao Yanli,Lin Gao.Modelling of crack propagation of gravity dams based on scaled boundary polygons.Journal of Hydraulic Engineering,2014,45(S1):30-37(in Chinese))
21陈楷,邹德高,孔宪京等.多边形比例边界有限单元非线性化方法及应用.浙江大学学报(工学版),2017,51(10):1996-2004(Chen Kai,Zou Degao,Kong Xianjing,et al.Novel nonlinear polygon scaled boundary finite element method and its application.Journal of Zhejiang University(Engineering Science),2017,51(10):1996-2004(in Chinese))
22章鹏,杜成斌,江守燕.比例边界有限元法求解裂纹面接触问题.力学学报,2017,49(6):1335-1347(Zhang Peng,Du Chengbin,Jiang Shouyan.Crack face contact problem analysis using the scaled boundary finite element method.Chinese Journal of Theoretical and Applied Mechanics,2017,49(6):1335-1347(in Chinese))
23 Natarajan S,Song C.Representation of singular fields without asymptotic enrichment in the extended finite element method.International Journal for Numerical Methods in Engineering,2013,96(13):813-841
24陈白斌,李建波,林皋.基于X-SBFEM的裂纹体非网格重剖分耦合模型研究.工程力学,2015,32(3):15-21(Chen Baibin,Li Jianbo,Lin Gao.Study on the coupling model of crack without remeshing based on X-SBFEM.Engineering Mechanics,2015,32(3):15-21(in Chinese))
25陈白斌,李建波,林皋.无需裂尖增强函数的扩展比例边界有限元法.水利学报,2015,46(4):489-496(Chen Baibin,Li Jianbo,Lin Gao.An extended scaled boundary finite element method without asymptotic enrichment of the crack tip.Journal of Hydraulic Engineering,2015,46(4):489-496,504(in Chinese))
26李建波,陈白斌,林皋.基于水平集算法的扩展比例边界有限元法研究.工程力学,2016,33(8):8-14(Li Jianbo,Chen Baibin,Lin Gao.Study on the extended scaled boundary finite element method based on level set algorithm.Engineering Mechanics,2016,33(8):8-14(in Chinese))
27傅兴安,李建波,林皋.基于X-SBFEM的非线性断裂数值模型研究.大连理工大学学报,2017,57(5):494-500(Fu Xing’an,Li Jianbo,Lin Gao.Study of numerical model of nonlinear fracture based on X-SBFEM.Journal of Dalian University of Technology,2017,57(5):494-500(in Chinese))
28 Jiang S,Du C.Coupled finite volume methods and extended finite element methods for the dynamic crack propagation modelling with the pressurized crack surfaces.Shock and Vibration,2017,2017:3751340
29 Jiang S,Du C.Study on dynamic interaction between crack and inclusion or void by using XFEM.Structural Engineering&Mechanics,2017,63(3):329-345
30 Fries TP.A corrected XFEM approximation without problems in blending elements.International Journal for Numerical Methods in Engineering,2008,75:503-532
31江守燕,杜成斌.一种XFEM断裂分析的裂尖单元新型改进函数.力学学报,2013,45(1):134-138(Jiang Shouyan,Du Chengbin.A novel enriched function of elements containing crack tip for fracture analysis in XFEM.Chinese Journal of Theoretical and Applied Mechanics,2013,45(1):134-138(in Chinese))
32 Pommier S,Gravouil A,Combescure A,et al.Extended Finite Element Method for Crack Propagation.ISTE Ltd and John Wiley&Sons,Inc.,2011:8-9
33 Mohammadi S.Extended Finite Element Method.Blackwell Publishing Ltd,2008:28-29
34 Sih GC,Paris PC,Erdogan F.Crack-tip stress intensity factors for plane extension and plane bending problems.Journal of Applied Mechanics,1962,29:306-312
35 Zamani A,Gracie R,Eslami MR.Cohesive and non-cohesive fracture by higher-order enrichment of XFEM.International Journal for Numerical Methods in Engineering,2012,90:452-483